1. Field of the Invention
The present invention relates to a laser beam scan optical system for writing image information with optical beams in digital copying machines, laser beam printers, optical disks, and the like. More particularly, the invention relates to a multi-beam scan optical system for writing image information with a plurality of laser beams.
2. Discussion of the Related Art
FIGS. 3A and 3B are explanatorydiagrams for an example of a conventional laser scan apparatus, which is used for the writing unit of a digital copying machine. Particularly, FIG. 3A is a plan view showing an optical system for a multi-beam scan, which simultaneously writes two lines on a photoreceptor with two laser beams. FIG. 3B is a side view showing the multi-beam scan optical system of FIG. 3A. The optical system includes a multi-beam laser diode array 01 as a semiconductor laser light source for simultaneously emitting two laser beams (see FIGS. 3A, 3B, and 4).
As shown in FIG. 4, the multi-beam laser diode array 01 includes an electrode substrate 02 and an LD (laser diode) chip 03 located on the electrode substrate 02. The LD chip 03 includes a pair of laser diodes LD.sub.1 and LD.sub.2 on a chip substrate 04. In those diodes, oscillation regions 06 and 07 are separated from each other by an insulation layer 05. The space between the laser diodes LD.sub.1 and LD.sub.2 (distance between the oscillation regions 06 and 07) is r.sub.1. Drive currents are fed from a drive circuit, not shown, through terminals Ta and Tb and electrodes 06a and 07a to the multi-beam laser diode array. Upon receipt of the drive currents, the laser diodes LD.sub.1 and LD.sub.2 emit first and second laser beams L.sub.1 and L.sub.2 forwardly, and back beams L.sub.1 ' and L.sub.2 ' backwardly. The diameters of the laser beams L.sub.1 and L.sub.2, at the time of their emission, are extremely small, 1 to 4 .mu.m in the direction parallel to the hetero interface. Accordingly, the figures are too small to define the laser beam diameters. For this reason, the size of them as a light source is expressed in terms of a divergence angle .theta..sub.1.
The multi-beam laser diode array 01 further includes a photo diode 08. The photo diode 08 receives the back beams L.sub.1 ' and L.sub.2 ' and outputs a light-quantity signal to a terminal Tc that is connected to a light-quantity controller, not shown.
As shown in FIGS. 3A and 3B, an optical system M for a multi-beam scan is located in the optical path for the laser beams L.sub.1 and L.sub.2 that are emitted from the oscillation regions 06 and 07. The optical system includes a collimate lens 010 of the focal distance f.sub.1, a cylindrical lens 012 of the focal distance f.sub.2, which has an optical power only in the subsidiary scan direction, a mirror 013, a rotating polygon mirror 014, an f-.theta. lens 015, and a cylindrical lens 016. The optical system M scans the surface (i.e., the photoreceptor surface) of a photoreceptor 018 in the form of a drum, which is rotated about the center shaft.
A first scan optical system M.sub.l consisting of the collimate lens 010 and the cylindrical lens 012 has the lateral magnification of m.sub.1 in the subsidiary scan direction. A second scan optical system M.sub.2 consisting of the f-.theta. lens 015 and the cylindrical lens 016 has the lateral magnification of m.sub.2 in the subsidiary scan direction. A photo sensor 019 is located near one of the ends of the photoreceptor 018. The output timing of an image signal applied to the laser diodes LD.sub.1 and LD.sub.2 is determined by a beam position detecting signal SOS (start of scan, see FIG. 3A) that is output from the photo sensor 019.
FIGS. 5 and 6 are explanatory diagrams for showing the spots of the laser beams L.sub.1 and L.sub.2 on the photoreceptor surface 018a when image information is written on the photoreceptor surface by using the optical system M. Here, a natural number r.sub.i, which represents the result of dividing the distance r.sub.3 of the scan lines along which the two adjacent laser beams L.sub.1 and L.sub.2 scan by the scan line width. (scan pitch) p, is defied as "interlaced scanning period".
The illustration of FIG. 5 shows a case where the number n of laser beams is 2, and the interlaced scanning period i is 1. The diameters of the beam spots a and h (as viewed in the subsidiary scan direction) of the laser beams L.sub.1 and L.sub.2 on the photoreceptor surface 018a are equal, d.sub.3. In the present specification, the diameter of the beam spot is defined as an areal part within which the maximum intensity of light, (1/e).sup.2 =0.135, extends. a.sub.1 and b.sub.1 indicate respectively the positions of the spots a and b in the first scan (indicated as the scan number (1) in the figures); a.sub.2 and b.sub.2, the positions of the spots in the second scan; a.sub.t and b.sub.t, the positions of the spots in the t-th scan (t=1, 2, 3, . . . ).
In FIG. 5, first and second scan lines are simultaneously traced with the first and second laser beams L.sub.1 and L.sub.2 in the first scan, or the scan of scan No. (1). Then, third and fourth scan lines are simultaneously traced with the first and second laser beams L.sub.1 and L.sub.2 in the scan of scan No. (2). Subsequently, the scans of scan Nos. (3), (4), (5), . . . are repetitively performed every two scan lines.
The illustration of FIG. 6 shows a case where the interlaced scanning period i is 5, that is, the distance r.sub.3 between the two laser beams L.sub.1 and L.sub.2 is 5p (p: scan line pitch). In this case, the second scan line and the fourth scan line are traced with the second laser beam L.sub.2 in the scans of scan Nos. (1) and (2). In the scan of scan No. (3) and the subsequent ones, the sixth, eighth, tenth, . . . scan lines are traced with the second laser beam L.sub.2. At the same time, the first, third, fifth scan lines, . . . , that are located preceding by five lines to those traced by the second laser beam L.sub.2, are simultaneously traced with the first laser beam L.sub.1.
In FIGS. 5 and 6, the outline of the conventional multi-beam scan apparatus was described by using the example where the two laser beams were used for the scanning operation. The multi-beam interlaced scanning operation as described above is allowed if the number n of laser beams and the interlaced scanning period i are mutually prime, viz., the number of the laser beams and the interlaced scanning period do not have any positive integers exactly divisible in common other than 1. In this respect, the multi-beam scan apparatus using three or more laser beams has been already known.
The conditions to realize the interlaced scanning will be described by using an example where four laser beams are used for the scan beams as shown in FIG. 8.
In the example, as shown in FIG. 8, an n (n=4) number of spots s.sub.j (j=1 to n) are arrayed at intervals i (i=3) times as large as the scan line interval p (viz., at interlaced scanning periods i). With the four beam spots thus arrayed, the scan is performed at subsidiary scan intervals q (q=4) times as large as the scan interval p. When the number t of scans reaches t.sub.0, t=t.sub.0 (t.sub.0 =3), one cycle is completed. In other words, the number t.sub.0 of one cycle is 3.
In FIG. 8, the scan number (No.) is denoted as t. The first scan is indicated by t=1; the second scan, by t=2; the third scan, by t=3; the fourth scan, by t=4; and so on. At the scan No. (number of scans), t=3, the scan of the first cycle is completed. The scans of t=4 to t=6 make up the scan of the second cycle. The scan lines of the first scan cycle are denoted as L.sub.1,0 to L.sub.1, 11, and the scan lines of the second scan cycle are denoted as L.sub.2,0 to L.sub.2, 11.
The spots s.sub.j (j=1 to 4) must be located on the scan lines, respectively. Hence, i and g are natural numbers.
To realize the interlaced scanning period, the following two conditions (a) and (b) must hold;
(a) All the scan lines are traced by scan. PA1 (b) The same scan line is not traced two times. PA1 t=-i+1, . . . -1, 0, 1, . . . , i-1 (integer representative of the scan number) PA1 k=1, 2, . . . , n (natural number representative of spot number) PA1 (1) The quantity of movement of the spot in the subsidiary scan direction for one main scan is n.multidot.p (n: the number of laser beams, and p: scan line interval). PA1 (2) i and n are natural numbers that are relatively prime (the greatest common divisor is 1). The interlaced scanning periods i and the number n of laser beams are tabulated in FIG. 10. As seen from the table of FIG. 10, the number n of laser beams is determined, the possible interlaced scanning periods i can be discretely obtained.
As shown in FIG. 8, in the scan of t=1 (first scan), four lines L.sub.1,0, L.sub.1,3, L.sub.1,6, and L.sub.1,9 are traced. The scans of t=1 to t=3 complete the first scan cycle. Accordingly, at t=4, the first scan in the next scan cycle starts. At this time, to satisfy the above two conditions (a) and (b), a spot S1 at t=4 must be at the position of a spot S5 as indicated by a dotted line in FIG. 8. In other words, when t.sub.0 =3, viz., the number of scans for one cycle is reached, the distance t.sub.0 .times.qp by which the spot has been moved must be equal to the length n.times.ip of the series of spots in one cycle. That is, EQU t.sub.0 q=ni (a)
Each spot interval i is filled with the t.sub.0 (t.sub.0 =3) number of scans. Therefore, ip=t.sub.0 p, and hence EQU i=t.sub.0 (b)
Substituting the equation (b) into the equation (a), we have EQU q=n (c)
As seen from FIG. 8, the position (line) of the spot S.sub.1 at the first scan of the second cycle, viz., at t=4, advances by ni line from the position (line) of the spot S.sub.1 at t=1. The quantity of the movement of a given spot S.sub.j for one cycle is the common multiple in of the interlaced scanning period i and the number n of spots.
Meanwhile, the condition (b) is that the same scan line is not repeatedly traced with each spot during one cycle. To satisfy the condition, the number n of spots and the interlaced scanning period i must be mutually prime. This will be described with reference to FIG. 9.
In FIG. 9, the scan line number L(t, k) can be described by using n of spots and the interlaced scanning period i and mathematically be expressed by EQU L.sub.(t, k) =(n.multidot.t+1)+i(k-1) (d)
where
In FIG. 9 and the equation (d), the number of the scan line scanned by a spot S.sub.k (k=1, 2, . . . ) for the scan number t is expressed by L(t, k). Accordingly, when t=0 and k=1, L(t, k)=1.
To avoid the repetitive scan of the same scan line by the spot, it is required that the scan line numbers of the scan lines scanned with two spots arbitrarily selected are not coincident with each other within the spot length ni. The requirement can be achieved when the following relations hold ##EQU2## where (t.sub.1 -t.sub.2) and (-k.sub.1 +k.sub.2) are arbitrary integers, and particularly EQU -n+1.ltoreq.(-k.sub.1 +k.sub.2).ltoreq.n-1-k.sub.1 +k.sub.2 .noteq.0(f)
In the above relations, t.sub.1 and t.sub.2 are two different scan numbers arbitrarily selected, k.sub.1 and k.sub.2 are different spot numbers also arbitrarily selected.
When (t.sub.1 -t.sub.2) and (-k.sub.1 +k.sub.2) have the same signs and are positive integers, if i and n are mutually prime, the least common multiple is i.multidot.n. From the equation (f), the right side (-k.sub.1 +k.sub.2) of the inequality (e) is less than i.multidot.n, i.e. (-k.sub.1 +k.sub.2)&lt;i.multidot.n. Therefore, the product of multiplying n by the arbitrary integer (t.sub.1 -t.sub.2) is not equal to i(-k.sub.1 +k.sub.2), which is an integer smaller than the least common multiple i.multidot.n of the numbers i and n. Hence, the inequality (e) holds.
Also for the integers of negative signs, the above inequality holds. When their signs are different, i.multidot.n is a positive integer. Therefore, the inequality (e) holds also in this case.
As seen from the foregoing discussion, the interlaced scanning is valid if the following two conditions are satisfied:
The factors to determine the picture quality of the reproduced picture are the spot diameter of the laser beam projected onto the photoreceptor surface, the width (i.e., scan pitch) of one scan line on the photoreceptor surface, laser power, and the like.
During a tone display by mesh-dots through the background exposure, when the diameter of the spot in the subsidiary scan direction is excessively large, the colored part in the highlight portion is too small to be reproduced into an image. In the shadow portion, the bleached part loses its contour. On the other hand, when the spot diameter is excessively small in the subsidiary scan direction, the colored part in the highlight portion is too large. Thus, the picture quality of the reproduced picture depends on the spot diameter of the laser beam on the photoreceptor surface.
In a case where in FIGS. 5 and 6, the diameter of the spot of the laser beam applied to the photoreceptor surface is d.sub.3 and the width (i.e., scan pitch) of one scan line on the photoreceptor surface is n, selection of the value K=(d.sub.3 /p) to be within the range between 1.4 to 1.8 will provide an excellent tone reproduction performance. This fact has been known. Reference is made to the paper by Tanaka in "Optics Four Academies, The 6th Color Optics Conference Paper Collection, p77, 1989".
Where the laser beam has a larger power, the value K is 1, viz., the spot diameter d.sub.3 of laser beam projected onto the photoreceptor surface is equal to the width (scan pitch) p of one scan line on the photoreceptor surface. Even in a state that the photoreceptor surface is scanned, with a plurality of scan beams, closely but without any overlapping of the beams thereon, an excellent image reproduction can be obtained. To realize a satisfactory image reproduction, K should be selected to be between 1.0 and 1.8, more preferably 1.4 and 1.8.
In the laser beam scan optical system using the multi-beam, the laser beams L.sub.1 and L.sub.2 pass through one and the same scan optical system. As to the distance r.sub.3 between the spots a and b of the laser beams L.sub.1 and L.sub.2 on the photoreceptor surface 018a and the diameter d.sub.3 of each of the spots a and b as viewed in the subsidiary scan direction, it is easy to set the distance r.sub.3 between the spots a and b of the laser beams L.sub.1 and L.sub.2 to be integer times as large as the scan pitch p, and to set the value K=(d.sub.3 /p) to be within 1.0 and 1.8, provided that the distance (as viewed in the subsidiary scan direction) between the laser diodes LD.sub.1 and LD.sub.2 of the laser diode array 1 as a light source and the diameter (as viewed in the subsidiary scan direction) of the oscillation region of each laser diode (spot diameter of the laser beam when it is emitted) are exactly obtained.
It is difficult to exactly obtain the beam spot diameter since the diameter is too small. The diameter d.sub.3 of each beam spot can accurately be obtained by using the divergence angle .theta..sub.1 of the laser beams L.sub.1 and L.sub.2 that are respectively emitted from the laser diodes LD.sub.1 and LD.sub.2. When the laser beam passes through the scan optical system, the peripheral part of the beam is removed by components making up the optical system, such as the lens, and the rotating polygon mirror. This peripheral removal of the laser beam is frequently called a "truncation". Because of this phenomenon, it is difficult to set the distance r.sub.3 between the spots a and b of the laser beams L.sub.1 and L.sub.2 to be interlaced scanning period times as large as the scan pitch p, and to set the value K=(d.sub.3 /p) to be within 1.0 and 1.8. For this reason, in the multi-beam scan optical system, it is difficult to obtain satisfactory tone reproduction performance and a good reproduction of the line image.